Triangularizability of trace-class operators with increasing spectrum

For any measurable set E of a measure space (X,μ), let PE be the (orthogonal) projection on the Hilbert space L2(X,μ) with the range ranPE={f∈L2(X,μ):f=0a.e. onEc} that is called a standard subspace of L2(X,μ). Let T be an operator on L2(X,μ) having increasing spectrum relative to standard compressions, that is, for any measurable sets E and F with E⊆F, the spectrum of the operator PET|ranPE is contained in the spectrum of the operator PFT|ranPF. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator T has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space (X,μ) is discrete or the operator T has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.